Formation of fluvial hanging valleys: Theory and...

Formation of fluvial hanging valleys:

Theory and simulation

Benjamin T. Crosby,1,2 Kelin X. Whipple,1,3 Nicole M. Gasparini,3,4

and Cameron W. Wobus1,5

Received 7 May 2006; revised 7 February 2007; accepted 30 March 2007; published 9 August 2007.

[1] Although only recently recognized, hanging tributary valleys in unglaciated,tectonically active landscapes are surprisingly common. Stream power–based riverincision models do not provide a viable mechanism for the formation of fluvial hangingvalleys. Thus these disequilibrium landforms present an opportunity to advance ourunderstanding of river incision processes. In this work, we demonstrate that thresholdsapparent in sediment flux–dependent bedrock incision rules provide mechanisms for theformation of hanging valleys in response to transient pulses of river incision. We simplifyrecently published river incision models in order to derive analytical solutions for theconditions required for hanging valley formation and use these results to guide numericallandscape evolution simulations. Analytical and numerical results demonstrate that duringthe response to base level fall, sediment flux–dependent incision rules may createeither temporary or permanent hanging valleys. These hanging valleys form as aconsequence of (1) rapid main stem incision oversteepening tributary junctions beyondsome threshold slope or (2) low tributary sediment flux response during the pulse ofmain stem incision, thus limiting the tributary’s capacity to keep pace with main stemincision. The distribution of permanent and temporary hanging valleys results from fourcompeting factors: the magnitude of base level fall, the upstream attenuation of theincision signal, the lag time of the sediment flux response, and the nonsystematic variationin tributary drainage areas within the stream network. The development of hangingvalleys in landscapes governed by sediment flux–dependent incision rules limits thetransmission of base level fall signals through the channel network, ultimately increasingbasin response time.

Citation: Crosby, B. T., K. X. Whipple, N. M. Gasparini, and C. W. Wobus (2007), Formation of fluvial hanging valleys: Theory and

simulation, J. Geophys. Res., 112, F03S10, doi:10.1029/2006JF000566.

1. Motivation

[2] When there is a change in the tectonic or climaticforcing on a landscape, hillslopes and channels adjust theirform until reestablishing equilibrium with the new boundaryconditions. The spatial distribution and response time of thistransient adjustment can exert a fundamental influence onthe growth and development of mountain ranges, the timingand delivery of sediment to depositional basins and otherfundamental processes in tectonically active landscapes. Asthis signal of adjustment propagates through fluvial sys-

tems, it can often be distinguished as an oversteepenedreach. This channel reach is referred to as a knickpoint anddifferentiates the relict, unadjusted portion of the landscapeupstream from the incised, adjusting reaches downstream(Figure 1). The origin and evolution of knickpoints hascaptured the interest of scientists for over a century [Gilbert,1896; Waldbauer, 1923; Penck, 1924; Davis, 1932; vonEngeln, 1940]. Though much of the recent work examiningknickpoints focuses on the transmission of incision signalsalong main stem channels [Hayakawa andMatsukura, 2003;Haviv et al., 2006; Frankel et al., 2007], others haveexamined the basin-wide distribution of knickpoints [Weisseland Seidl, 1998; Bishop et al., 2005; Bigi et al., 2006;Crosbyand Whipple, 2006; Berlin and Anderson, 2007]. Some ofthese studies recognize that in incising, nonglacial, tecton-ically active landscapes, knickpoints are commonly locatedat the junction between tributaries and the incised trunkstream [Snyder et al., 1999; Crosby and Whipple, 2006;Wobus et al., 2006]. These knickpoints keep the tributaryelevated or ‘hung’ above the trunk stream. In the presentstudy, we explore potential mechanisms for the formation ofthese hanging valleys.

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1Department of Earth, Atmospheric and Planetary Sciences, Massachu-setts Institute of Technology, Cambridge, Massachusetts, USA.

2Now at Department of Geosciences, Idaho State University, Pocatello,Idaho, USA.

3Now at School of Earth and Space Exploration, Arizona StateUniversity, Tempe, Arizona, USA.

4Department of Geology and Geophysics, Yale University, New Haven,Connecticut, USA.

5Now at Cooperative Institute for Research in Environmental Sciences,University of Colorado, Boulder, Colorado, USA.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2006JF000566$09.00

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http://dx.doi.org/10.1029/2006JF000566

[3] The Waipaoa River on the North Island of NewZealand has provided an excellent location to study knick-point distribution within fluvial basins because the age ofdisturbance is well established [Berryman et al., 2000; Edenet al., 2001] and there are an abundance of transient land-forms such as terraces, incised inner gorges and knickpoints[Crosby and Whipple, 2006]. Even in this ideal transientlandscape we found it difficult to definitively discernbetween two proposed mechanisms for the formation ofhanging valleys [Crosby and Whipple, 2006]. The firstproposed mechanism hypothesized that discrete knickpointsmigrating up the trunk streams established hanging valleysas the discrete main stem knickpoint passed tributaryjunctions. The second mechanism proposed that the mainstem incised gradually, never containing a distinct knick-point, but this progressive incision was fast enough tooutpace tributary adjustment. The presence and persistenceof knickpoints at tributary junctions limits the upstreamcommunication of subsequent signals of base level change

into the upper portions of the channel network, thusextending basin response time following disturbance.[4] The most broadly utilized formulation for fluvial

bedrock incision, the detachment-limited stream powermodel [Howard and Kerby, 1983; Howard, 1994; Whippleand Tucker, 1999] (henceforth simply termed the streampower model), does not predict the formation of hangingtributaries [e.g., Niemann et al., 2001]. This discrepancybetween field observation and model behavior suggests aclear inadequacy in standard stream power river incisionmodels. Recently developed sediment flux–dependent bed-rock incision relations allow sediment to behave either as atool for incising the bed or as armor, inhibiting erosion[Sklar and Dietrich, 1998; Whipple and Tucker, 2002;Parker, 2004; Sklar and Dietrich, 2004; Gasparini et al.,2006]. We find that these new relations provide mecha-nisms for explaining the formation and persistence ofhanging tributaries at threshold drainage areas [Crosbyand Whipple, 2006; Wobus et al., 2006; Gasparini et al.,2007]. Indeed, the prevalence of hanging tributaries in

Figure 1. Landscapes responding to a discrete period of incision often possess hanging tributaries orsubcatchments isolated above the main stem by a large step in channel elevation. In (a) this topographicrendering of a portion from the Waipaoa River on the North Island of New Zealand, (b) channels drainingthe outlined tributary basins are elevated above and segregated from the trunk streams by steepknickpoints. In order to understand how these features develop during a transient pulse of incision, weutilize analytical and numerical methods to model (c) the response of channel profiles and (d) drainagebasins to discrete base level fall events. Figures 1a and 1b are modified from Crosby and Whipple [2006].

F03S10 CROSBY ET AL.: FORMATION OF FLUVIAL HANGING VALLEYS

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tool-starved environments may provide the strongest exist-ing field evidence supporting sediment flux–dependentincision rules. Important tasks undertaken here include(1) the quantitative assessment of the conditions underwhich sediment flux–dependent incision models predicthanging valley formation, and (2) careful determination ofwhat characteristics of these incision models are required byour field observations.

2. Approach and Scope

[5] In this work, we utilize analytical and numericalmodels to explore whether thresholds in existing relationsfor stream incision by sediment abrasion provide a plausiblemechanism for the formation of hanging valleys. Thetheoretically demonstrated consequences of these thresholdsare then considered relative to existing field studies of thedistribution of hanging valleys [Weissel and Seidl, 1998;Bishop et al., 2005; Crosby and Whipple, 2006; Wobus etal., 2006]. Although plausible alternative explanations forthe formation of hanging tributaries exist, thresholds insediment flux–dependent erosion relations provide anexcellent opportunity to compare theoretical predictionsagainst field data.[6] This study recognizes that although it is difficult to

distinguish between steady state channel profiles predictedby different stream incision rules, each incision rule dem-onstrates a unique behavior during its transient response tosome disturbance [Howard and Kerby, 1983; Stock andMontgomery, 1999; Whipple et al., 2000; Whipple andTucker, 2002; van der Beek and Bishop, 2003]. To evaluatethe real-world applicability of any particular incision rule, itthus becomes necessary to compare the predicted transientresponse with field observations from a disequilibriumlandscape with a known age and type of disturbance. The

direct comparison between modeled and observed transientlandscapes thus provides an excellent opportunity to recog-nize the strengths and weaknesses of the present formula-tions for stream incision (Figure 1).[7] In this paper, we provide a comparative analysis of

how four different stream incision rules respond to twodifferent scenarios for base level fall. Two of the fourincision rules are simplified versions of two recent sedimentflux–dependent incision rules [Sklar and Dietrich, 1998;Parker, 2004; Sklar and Dietrich, 2004; Gasparini et al.,2006; Gasparini et al., 2007]. For reference, we also modelthe channel response to base level fall using the detachment-limited stream power incision rule [e.g., Howard and Kerby,1983;Whipple and Tucker, 1999] and a simplified transport-limited incision rule [Willgoose et al., 1991; Paola et al.,1992a; Tucker and Bras, 1998].[8] Because the incision rate in three of the four studied

incision rules is directly dependent on sediment flux, wefind it advantageous to employ CHILD [Tucker et al.,2001a, 2001b], a two-dimensional landscape evolutionmodel (Figure 1d) wherein changes in sediment productionand transport capacity are explicitly accounted for duringthe transient response [Gasparini et al., 2006, 2007]. In thisstudy we use a landscape evolution model to examine theinteraction between an incising trunk stream and tributarychannels with a range of drainage areas.[9] The two base level fall scenarios modeled here

represent end-members for the range of forcing an incisingriver may experience in response to a sudden, but finitepulse of base level fall. Gasparini et al. [2007] considersprimarily main stem response to a sustained increase in therate of rock uplift (or base level fall), as opposed to the finitebase level fall events considered here. Our two base levelfall scenarios help evaluate the relative sensitivities of thefour incision models to the type of base level fall and allowus to mimic the behavior of smaller tributary networksnested within a larger catchment. In the first scenario, wesubject the modeled catchment to an instantaneous drop inbase level at its outlet. In order to provide comparison totypical field settings (including the Waipaoa River basin,introduced earlier), we elect to drop base level 50 m or �1/3of the modeled basin’s steady state fluvial relief. Forexample, instantaneous base level fall could result fromstream capture or surface rupture along a fault [e.g., Sklar etal., 2005]. In the second scenario, we examine the responseof the river network following a finite but prolonged periodof base level fall. In this scenario, we impose a 10 foldincrease in the rate of base level fall and allow the model torun until the accumulated base level fall is equivalent to themagnitude of the instantaneous drop in base level (50 m,Table 1). This type of disturbance could result from sealevel fall over the shelf slope break [e.g., Snyder et al.,2002] or a temporary increase in rock uplift rate. Thisscenario also serves to simulate the base level fall signal asubcatchment nested within a larger basin might experienceif the outlet to the larger basin was subjected to aninstantaneous base level fall.[10] We begin our analysis by introducing the four stream

incision rules utilized here and discussing the simplifica-tions we made to these rules that allow us to write analyticalsolutions. We then outline the dependence of each model’sincision rate on local stream gradient and discuss the

Table 1. Parameters Used During Model Runs

Parameter Value

Basin dimensions, m 10000 � 10000Basin area, m2 1.0 � 108

Node spacing, m 100Background rate of base level fall, U, m/yr 1.00 � 10�3

KD,a,b,c,d m2/yr 5, 5, 5, 4

Instantaneous base level fall,e m 50Progressive base level fall,f m 50mt 1.5nt 1Kt, m

3�2mt/yr 2.00 � 10�5

KSP, m�(2m+1) 4.00 � 10�5

KSA, m�0.5 5.00 � 10�2

KGA, m�1 7.00 � 10�3

mb,c,d 0.5, �0.25, 0nb,c,d 1, �0.5, 0kw, m

1�3b/yrb 1b 0.5kq, m

3�2c/yr 1c 1b 0.5

aTransport-limited model run.bDetachment-limited stream power model run.cSaltation-abrasion model run.dGeneralized abrasion model run.eDiscrete elevation change basin outlet.fIncrease rate of base level fall 10� and run model until 50 m new of

material is exhumed/uplifted.


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presence or absence of theoretically predicted instabilities.After a brief discussion of the utility and mechanics of theCHILD model, we present, for each of the four incisionrules, the transient response of the trunk and tributariesfollowing the two base level fall scenarios. Our subsequentdiscussion focuses on the interaction between trunk streamincision and the development of hanging tributary valleys.

3. Stream Incision Rules

[11] Our primary motivation is to understand the transientresponse to base level fall of channels governed by sedimentflux–dependent bedrock incision rules. For comparison, wealso provide an analysis of the transient response to baselevel fall of channels governed by the detachment-limitedstream power incision rule and a simplified transport-limited stream incision rule. This comparative analysisemphasizes the unique attributes and sensitivity of thesediment flux–dependent incision models. In the followingsection, we introduce all four of the stream incision rulesemployed in the CHILD landscape evolution model.[12] In all cases, we model the potential development of

extremely steep channel reaches. Under these conditions,the small-angle approximation, (where sin(a) � tan(a) anda denotes the bed angle in radians), is no longer valid. Inthis analysis, we take considerable care to avoid the small-angle approximation. In all equations in this text, thevariable S represents sin(a), not channel gradient. In allfigures, however, we transform sin(a) to the more intuitivevariable, channel gradient (tan(a)), using the relation,Gradient = tan(arcsin(S)).

3.1. Transport-Limited and Stream Power IncisionRules

[13] We only briefly discuss the transient behaviors of thetransport-limited and the detachment-limited stream powerincision rules as these have been well explored by otherworkers [e.g., Howard and Kerby, 1983; Willgoose et al.,1991; Whipple and Tucker, 1999, 2002]. In this analysis, weuse a simplified rule for transport-limited incision where thevolumetric sediment transport capacity is a power lawfunction of unit stream power. This model follows thegeneral form of many other sediment transport equations[Meyer-Peter and Mueller, 1948; Wilson, 1966; FernandezLuque and van Beek, 1976] where the volumetric sedimenttransport capacity, Qt is a power law function of the stream’sexcess shear stress:

Qt / W t � tcð Þ3=2; ð1Þ

where W is channel width, t is the basal shear stress and tcis the critical shear stress. All calculations of channel widthin this paper (both in analytical and numerical analysis)utilize a power law relation between width and discharge,W = kwQ

b, and a power law relationship between waterdischarge and drainage area, Q = kqA

c. Combined, these twoequations describe the relation between width and drainagearea as W = kwkq

bAbc. Assuming uniform, steady flow in awide channel and utilizing the Darcy-Weisbach flow

resistance equation [e.g., Tucker and Slingerland, 1996],shear stress can be expressed as a power law function ofupstream drainage area, A, and the sine of the bed angle, S:

t ¼ ktA1=3S2=3; ð2Þ

where kt is a constant term characterizing fluid properties,bed morphology and basin geometry. As detailed inprevious work [Willgoose et al., 1991; Paola et al.,1992b; Tucker and Slingerland, 1997; Gasparini et al.,2007], we assume a negligible threshold of motion for thefloods of interest (t tc) and substitute equation (2) inequation (1), allowing us to write Qt as a power lawfunction of the upstream drainage area and the sine of thebed angle, Qt / kwkt

3/2A1S1, or in a generalized form:

Qt ¼ KtAmt Snt ; ð3Þ

where Kt is a dimensional coefficient describing thetransportability of the channel sediment and mt and nt aredimensionless positive constants. Although a critical shearstress could easily be added to numerical simulations (andwould importantly influence equilibrium channel slope atlow rates of relative base level fall), we retain this simplifiedform in order to derive analytical solutions that provideconsiderable insight into the problem addressed here: how,why, and where fluvial hanging valleys form. Under theconditions that lead to hanging valley formation, theassumption that t tc is reasonable. We hold nt = 1 inall simulations, consistent with most relations for bed loadtransport where Qt is proportional to basal shear stress to the3/2 power [e.g., Whipple and Tucker, 2002, equation (5)].Given that the change in elevation is a consequence of thedifference between the rate of base level fall, U, and thedownstream divergence of sediment flux, we can solve forhow incision rate varies as a function of S or sin(a),

dz

dt¼ U � 1

1� lp

� �d

dx

1

WKtA

mt Sntð Þ� �

; ð4Þ

where lp is the sediment porosity and W is the channelwidth. The effects of a shear stress threshold on channelconcavity can be mimicked by increasing the value for mt

and appropriately adjusting Kt, as first proposed by Howard[1980]; mt could reasonably vary between 1.0 and 1.5, andin all of our experiments we set mt = 1.5 (Table 1). Giventhe form of the incision relation formulated above and nt =1, the incision rate at a particular drainage area is a linearfunction of the sine of the bed angle (Figure 2 and Table 1).Because this incision relation has the form of a nonlineardiffusion equation, an oversteepened reach created by baselevel fall is expected to decay rapidly as it propagatesupstream.[14] The stream power incision rule assumes that trans-

port capacity well exceeds the imposed sediment load andthus the rate of channel incision is limited simply by thechannel’s capacity to detach bedrock from the channel bed[e.g., Howard and Kerby, 1983; Whipple and Tucker, 1999].In this relation, there is no explicit functional dependence ofthe incision rate on the sediment flux. Changes in bed


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elevation are determined by differencing the backgroundrate of base level fall, U, and the bedrock incision rate. Thebedrock incision rate is a power law function of A, theupstream drainage area, and S, the sine of the bed angle:

dz

dt¼ U � KSPA

mSn; ð5Þ

where KSP is a dimensional coefficient describing theerodibility of the channel bed as a function of rock strength,bed roughness and climate, and m and n are dimensionlesspositive constants.[15] This relation can be rewritten into the same functional

form as used to describe kinematic waves [Rosenbloom andAnderson, 1994]. As a consequence, base level fall signalsmodeled with the stream power incision rule propagateupstream through the network as discrete waves. The formof the oversteepened reach may evolve as it propagatesupstream depending on the value of n [Weissel and Seidl,1998; Tucker andWhipple, 2002]. In this work we utilize unitstream power and assume m = 0.5 and n = 1 [e.g., Whippleand Tucker, 1999]. This results in waves that propagate

upstream without changing form and as with the transport-limited model, defines a linear dependence of the incisionrate (at any given drainage area) on sin(a) (Figure 2).

3.2. Sediment Flux–Dependent Models for ChannelIncision

[16] Sediment flux–dependent incision rules introducedby Sklar and Dietrich [1998, 2004], Whipple and Tucker[2002] and Parker [2004] explicitly model the dual role ofsediment in bedrock channel incision. In these incisionrules, when sediment flux exceeds transport capacity, sed-iment covers and armors the bed against incision and forcesthe channel toward a transport-limited behavior. For lowvolumes of sediment flux, insufficient tools are available toimpact and abrade the channel bed and results in conditionsthat approach the detachment-limited state.[17] In this analysis, we distinguish two classes of sedi-

ment flux–dependent incision rules: one where saltationdynamics (on a planar bed) plays a fundamental role inchannel incision and another where although incision issimilarly accomplished by abrasion alone, saltation dynam-ics are not explicitly modeled. These models are derived

Figure 2. We demonstrate the dependence of incision rate on channel gradient for four channel incisionrules. Each line is for a fixed drainage area (1 � 106 m2) and sediment load. All calculations are madewithout applying the small-angle approximation. Instead of plotting the x axis with sin(a), we transformthis term to channel gradient, tan(arcsin(sin(a))). The parameters for each of the models are the same asthose used in the numerical simulations presented later in the paper. As this figure illustrates, theparameters for each of the models were chosen in order to produce similar steady state slopes (Sss.). Forboth the stream power and the transport-limited models, incision rate increases rapidly with gradients. Inthe saltation-abrasion model, incision rate initially increases with increasing gradient, exceeding thebackground rate of base level fall at Sss. After reaching a maximum at Speak, the incision ratemonotonically decreases with increasing gradient, eventually dropping below the background base levelfall rate (the gray horizontal line) at Shang. The incision rate in the generalized abrasion model initiallyincreases rapidly with gradient and levels off as it asymptotically approaches a maximum incision rate atImax.


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through minor simplifications of existing sediment flux–dependent river incision models proposed by Sklar andDietrich [1998, 2004] and Parker [2004]. Our simplifica-tions facilitate analytical exploration and provide directcomparison to the behavior of previously discussed incisionrules [e.g., Gasparini et al., 2006]. Our simplified formu-lations very closely approximate the behavior of the originalequations.3.2.1. Saltation-Abrasion Incision Rule[18] The incision rule developed by Sklar and Dietrich

[1998, 2004] provides a process-specific, mechanistic rela-tion for bedrock incision by the abrasion of saltating bedload of a single grain size on a planar bed. In their rule, theincision rate is an explicit function of both the flux ofkinetic energy normal to the bed (impacts) and the fractionof the bed exposed to those impacts. The incision rate forthis model, ISA, is written as the product of three measurableterms [Sklar and Dietrich, 2004, equation (1)]: the volumeof rock prepared for transport per particle impact, Vi, the rateof particle impacts per unit area, per unit time, Pr, and thefraction of the channel’s bedrock bed exposed to incision,Fe:

ISA ¼ ViPrFe: ð6Þ

Each term in this expression can be expanded and expressedas a function of excess shear stress. Sklar and Dietrich[2004, equation (24a)] combine the three terms above into asimple expression for bedrock incision. In this analysis, weuse their equation (24b), applicable at moderate transportstages (1 < t*/t*c < 10) in which the term representing theinfluence of incipient suspension is removed. We write theirequation (24b) in a form analogous to their equation (24a):

ISA ¼ Rbg

25ev

� � Qs

W1� Qs

Qt

� �� t*

t*c� 1

!�0:5224

35; ð7Þ

where Rb is nondimensional buoyant density of thesediment, g is gravitational acceleration, ev is the energyrequired to erode a unit volume of rock, Qs is volumetricsediment flux, W is channel width, Qt is volumetricsediment transport capacity, t* is nondimensional shearstress and t*c is nondimensional critical shear stress.Equation (7) only applies to net erosional settings whereQs is less than or equal to Qt. The dual role of sediment fluxin determining incision rate is apparent in the second termof equation (7) where as the sediment flux term, Qs, goes tozero, so does ISA. In contrast, as Qs approaches the transportcapacity, Qt, then incision rate again goes to zero as thechannel bed becomes fully covered with sediment. Duringthe transient response, sediment flux can vary dramaticallythrough time and throughout the basin as hillslopes andtributaries respond to changes in the elevation of the trunkstream. The negative exponent on the third term, excesstransport stage, is a consequence of the explicit inclusion ofsaltation dynamics and determines that incision ratedecreases with increasing excess bed shear stress, all elseheld equal. Using linear regression analysis, we find that theexcess transport stage can be reasonably approximated as apower law function of dimensional shear stress and criticalshear stress. In order to allow an analytical solution, we

further approximate the shear stress exponent as �0.75rather than the �0.88 value found through this empiricalregression analysis:

t*

t*c� 1

!�0:52

� 2t*

t*c

!�0:88

� 2ttc

� ��0:88

� 2ttc

� ��0:75

: ð8Þ

We find that this approximated form of the excess transportstage term provides a reasonably strong fit to their empiricaldata as well. Substituting the final term from thisapproximation back into the third term in equation (7), wecan rewrite that expression as

ISA ¼ 2Rbgt0:75c

25ev

� � Qs

W1� Qs

Qt

� �� tð Þ�0:75: ð9Þ

We then substitute equation (2) for the shear stress term andmove the kt term into the first bracketed term inequation (9). In this final expression, which we genericallyrefer to as the saltation-abrasion incision rule, we group theterms in the first bracket of equation (9) as constants thatcharacterize the channel’s erodibility, KSA. Following[Whipple and Tucker, 2002], we also define the two termsin the second bracket as f (Qs), an expression reflecting therole of sediment flux in setting the incision rate:

f Qsð Þ ¼ Qs

W1� Qs

Qt

� �: ð10Þ

The first term in this second bracket, (Qs/W), quantifies thevolume of sediment per unit width available as tools. Thevalue for the Qs term is the sediment supply at a particularpoint in time and space. Under equilibrium conditions, Qs isequal to the upstream drainage area times the rate of baselevel fall (or rock uplift) times the fraction of sedimentdelivered to the channel as bed load (b). As discussed earlier,width is assumed to be a power law function of drainage area(see section 3.1). The second term, (1 � (Qs/Qt)), describesthe cover effect, where increasing sediment flux relative totransport capacity diminishes the incision rate. The form ofthe transport capacity is defined above in equation (3).When the sediment flux surpasses the transport capacity,sediment is deposited and equations (9) and (10) are nolonger physically valid. The final expression for oursaltation-abrasion rule shares the same functional form asthe generic sediment flux–dependent incision rule pre-sented by Whipple and Tucker [2002, equation (1)], butinstead uses significantly different values for the exponentson drainage area and slope:

ISA ¼ KSA f Qsð ÞA�1=4S�1=2: ð11Þ

As we will address in subsequent sections, the negativeexponent on S, the sine of the bed angle, exerts an importantinfluence on the responses of tributaries during a transientperiod of incision and reflects the explicit inclusion ofsaltation dynamics over a planar bed. When addressing therelationship between incision rate and channel gradient(Figure 2), it is important to recognize that equation (11) hastwo gradient-dependent terms: first the Qt term within f (Qs)is dependent on the sine of the bed angle (see equations (3)


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and (10)) and second, the sine of the bed angle term, S, withthe negative exponent that occurs following the substitutionfor shear stress (equation (2) into (9)). As Figure 2demonstrates, we find that the incision rate increases withincreasing channel gradient to a maximum and thendecreases with ever-increasing gradients.[19] The steady state channel profiles, slope-area rela-

tions, and dependence of incision rate on channel gradientfor this simplified expression are almost identical to thosepredicted by the original Sklar and Dietrich [2004,equation (24a)] incision rule. The gradient of the channelat large drainage area asymptotically approaches that pre-dicted by the purely transport-limited model. The steadystate longitudinal form of the channel deviates from thetransport-limited gradient only at small drainage areas inupper reaches of the channel network [Sklar and Dietrich,2006; Gasparini et al., 2007, Figure 1a]. However, understeady state conditions, there is a critical drainage areabelow which channel gradients become infinite. This criticalarea, as described in detail by Gasparini et al. [2007,equation (27)], is a consequence of two characteristics ofsmall drainage areas: (1) low sediment supply and (2) highgradients. Because incision depends on saltating bed load,these two factors limit the interaction of sediment with thebed, thus limiting incision and the applicability of thismodel at extremely small drainage areas. It should also benoted that decreasing channel width at low drainage areasmay offset the diminishing sediment flux, therefore allow-ing the channel to maintain a constant sediment flux per unitwidth. Regardless, the steeper slopes observed at lowdrainage areas (due to the need to transport sediment atshallow flow depths) result in longer hop lengths forsaltating grains and a decrease in the efficiency of incisionby bed load abrasion. In the CHILD model, diffusivehillslope processes are responsible for sediment productionand transport at drainage areas less than this critical area.We reserve further discussion of the dependence of incisionrate on gradient for the proceeding section on modelinstabilities.3.2.2. Generalized Abrasion Incision Rule[20] Parker [2004] presents an incision model in which

two processes dominate erosion of the channel bed: pluck-ing of bedrock blocks and abrasion by saltating bed load.For the abrasion component, the Parker model employs asimplified version of the incision model of Sklar andDietrich [2004], based on analogy to a model of down-stream fining due to a constant coefficient of wear. Thissimplification was not intended to correct the model ofSklar and Dietrich, but instead to provide comparison withother mechanisms of incision. These comments notwith-standing, the Parker model applied to bed load abrasionalone can be used to highlight the role of specific terms inthe formulation of Sklar and Dietrich [2004] with respect tothe formation of hanging valleys. With this in mind, wefocus only on the bed load abrasion process (same methodapplied by Gasparini et al. [2007]). This generalizedabrasion (GA) rule facilitates simpler analytical solutionsand a more direct comparison between the incision rules ofParker [2004] and Sklar and Dietrich [2004]. The onlyfunctional difference between this generalized abrasion ruleand the saltation-abrasion model (equation (11)) is that theexponents on both the drainage area and the sine of the bed

angle are equal to zero (recall that the negative exponents inthe saltation-abrasion rule reflected details of saltationdynamics over a plane bed that have been dropped in theformulation of this generalized abrasion model):

IGA ¼ KGA

Qs

W1� Qs

Qt

� �; ð12Þ

where KGA is a dimensional constant equal to (r/Ls)where r is the fraction of the particle volume detached offthe bed with each collision and Ls is saltation hop length.Note that in order to have a constant incision rate, IGA, atsteady state, the f (Qs) term in equation (12) (everything butthe KGA) must be constant. Figure 2 shows that incision ratein this generalized abrasion rule monotonically increasestoward a maximum incision rate with increasing values ofchannel gradient. This behavior is a consequence of thegradient (or sine of the bed angle) term in the equation forthe volumetric transport capacity (equation (3)). Like thesaltation-abrasion incision rule, the generalized abrasionincision rule predicts that at large drainage areas, thechannel gradient is determined by the system’s sedimenttransport capacity. At steady state the generalized abrasionincision rule also predicts a critical drainage area at whichchannel gradients become infinite as a consequence of lowsediment fluxes [Gasparini et al., 2007, equation (32)]. Asin the saltation-abrasion incision rule, the generalizedabrasion rule’s threshold drainage area increases withdecreasing values of the dimensional constant, KGA

(stronger rocks). Unlike the saltation-abrasion incisionrule, the generalized abrasion rule’s critical drainage areais insensitive to rate of base level fall (or rock uplift). Wereserve further discussion of the dependence of incisionrate on gradient for the proceeding section on modelinstabilities.

4. Predicted Transient Instabilities

[21] There are no transient instabilities in the transport-limited or the stream power incision rules as formulatedabove. The proceeding section outlines instabilities pre-dicted to occur in the two sediment flux–dependent incisionrules. These instabilities provide mechanisms for the for-mation of temporary and permanent hanging valleys. Ineach case, we assume that during the initial response to baselevel fall, changes in sediment flux lag behind the profileadjustment (or Qs = Qs�initial). We justify this assumption bysuggesting that the hillslope response (which determinessediment flux) is dependent on the transmission of theincision signal through the network. If this assumption isviolated and sediment flux does not lag during adjustment,the instabilities predicted below will underestimate thresh-old gradients or overestimate threshold drainage areas.

4.1. Instabilities in the Saltation-Abrasion IncisionRule

[22] In Figure 2, we demonstrate that the steady stateincision rate predicted by the saltation-abrasion incision rulederived from Sklar and Dietrich [1998, 2004] initiallyincreases from extremely small values at low gradients,past the steady state gradient at Sss, to a maximum incisionrate at Speak. For gradients greater than Speak, the incision


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rate decreases monotonically with increasing gradient,reaching an incision rate equal to the background rate ofbase level fall (or rock uplift) rate at Shang. The nonmono-tonic form of the solid curve in Figure 2 is the basis for theinstabilities observed in the saltation-abrasion incision rule.[23] The first of the two instabilities in the saltation-

abrasion incision rule occurs if a channel’s gradient exceedsthe gradient required for the incision rate to keep pace withthe background base level fall rate, Shang (Figure 2). Beyondthis critical gradient, the stream can no longer effectivelyerode the bed at a rate sufficient to keep pace with thebackground base level fall rate, U. This creates a runawaynegative feedback that results in the formation of a perma-nent hanging valley (a waterfall) in all cases.[24] Both steady state channel gradient, Sss, and Shang are

recognized as two of the three roots of equation (11) whenISA = U, where U is the rate of base level fall (Figure 2). Wecan derive analytical expressions for Sss and Shang bystarting with equation (11) and setting ISA = U, substitutingequation (3) for Qt and setting Qs = bAU (the steady statesediment flux at a given base level fall rate, where b is thepercentage of the eroded material that is transported as bedload). We also use a power law width-discharge relation anda power law discharge-area relation to set W = kwkq

bAbc asoutlined by Whipple and Tucker [1999]:

U ¼ KSA

bAUkwkbqA

bc1� bAU

KtAmt Snt

� �" #A�1=4S�1=2: ð13Þ

The solution for steady state gradients as a function of areais found by setting nt = 1 (as discussed above), making thesubstitution x = S�1/2 thus allowing equation (13) to berearranged into a relatively simple cubic form,

x3 � S�1t x ¼ �S�1

t

1

K 0bA1�bc�1=4

� �; ð14Þ

where we group variables and define St as

St ¼bUKt

A1�mt : ð15Þ

In the two expressions above, St defines the steady statetransport-limited gradient [e.g., Whipple and Tucker, 2002],K0 = KSA/(kwkq

b), and the generic cubic form of equation (14)is x3 + px = j. The three real roots of this cubic equation are

S1 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 1

3p

� �scos

qþ 2p3

� � !�2

ð16Þ

Sss ¼ 2


3p

� �scos

q3

� � !�2

ð17Þ

Shang ¼ 2


3p

� �scos

qþ 4p3

� � !�2

; ð18Þ

where q is defined as

q ¼ cos�1

1

2jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� 1

27p3

r0BB@

1CCA: ð19Þ

These roots define the three potential values for sin(a) atwhich the incision rate is equivalent to the background rateof base level fall (Figure 2). The first root, S1, is unphysicaland is not addressed in Figure 2. The second root, Sss, is aphysically meaningful solution that describes the value ofsin(a) at steady state. At this value, the channel exhibits astable, partially covered bed and can erode and transportmaterial at a rate sufficient to keep pace with thebackground rate of base level fall (or rock uplift). Thethird root, Shang, is often, but not always, also physicallyvalid and describes the value of sin(a) above which theincision rate decreases below the background base level fallrate (Figure 2), facilitating the formation of a permanenthanging valley. Naturally, Shang is only physically mean-ingful for values between 0 and 1 (sin(a) = 1 for a verticalcliff). At values of sin(a) higher than Shang, the saltating bedload no longer impacts the bed effectively or frequentlyenough to maintain a sufficiently high incision rate to keeppace with the background rate of base level fall. Basinsupstream of these oversteepened reaches are terminallydivorced from the lower reaches unless other processes actto reduce channel gradient. Combining equations (14), (15)and (19) into (18) we can derive the value of sin(a) at Shangas

Shang ¼ 2

ffiffiffiffiffiffiffi1

3St

r cos 1

3cos�1

�1

2StK 0bA1�bc�1=4ffiffiffiffiffiffiffiffiffiffi1

27S3t

s0BBBB@

1CCCCAþ 4

3p

0BBBB@

1CCCCA

266664

377775

�2

:

ð20Þ

As presented in Figure 3, the channel gradient required toform a permanent hanging valley increases with drainagearea (the increase in Shang with drainage area is exactlycommensurate with the decrease in Sss with drainage area).In large tributaries, there is no physically meaningfulsolution for Shang, because values predicted by equation (20)exceed unity. As discussed later, this implies that largertributaries will be able to keep up with a given pulse of mainstem incision but smaller tributaries experiencing the samemagnitude base level fall signal at their junction with themain stem might not. This determines which tributariesform hanging valleys during periods of rapid base level fall.[25] The second transient instability associated with the

saltation-abrasion incision model creates temporary hangingvalleys that fail to keep pace with main stem incision for aperiod of time, but eventually recover and equilibrate to themain stem. These occur when the transient pulse of incisionincreases the value of sin(a) greater than Speak, but less thanthe previously discussed Shang (Figure 2). We can derive thevalue for Speak at any particular drainage area by solving forwhen the change in incision rate with respect to sin(a)


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equals zero. We find that this value is three times the initialtransport-limited channel gradient or

Speak ¼3bKt

UinitialA1�mt : ð21Þ

Because mt is greater than one (Table 1), the negativeexponent on area in equation (21) means that Speak increaseswith decreasing drainage area. This implies that thedifference between Speak and Shang decreases with decreas-ing drainage area, as does the difference between Sss andShang (Figure 3). As a result, smaller tributaries are far moresusceptible to the formation of permanent hangs than largertributaries which will either (1) respond quickly in anessentially transport-limited manner or (2) develop atemporary hanging valley and then recover. For aninstantaneous pulse of base level fall, a temporary hangingvalley forms as the pulse of main stem incision steepens thetributary outlet to values between Speak and Shang. The

gradual subsequent incision of this oversteepened reachexceeds the background rate of base level fall, and thuseventually decreases the height and maximum gradient ofthe oversteepened reach, resulting in a positive feedbackloop in which gentler gradients promote greater incisionrates and thus more rapid decay of the temporary hangingvalley knickpoint. This positive feedback leads to theeventual recovery of the temporary hanging valley to agraded condition. Note that we discuss here only finiteepisodes of increased base level fall. If accelerated rates ofmain stem incision were sustained indefinitely owing to apersistent change in rate of base level fall (or rock uplift),permanent hangs would from wherever channels aresteepened beyond Speak – this may be the most commoncircumstance leading to formation of hanging tributaries[Wobus et al., 2006], but is not the case in the Waipaoa fieldsite. In addition, our derivations of Speak and Shang assumewe are observing the initial response of the channel (whereQs = Qs�initial); any partial communication of the incisionsignal to the upper basin will increase sediment delivery and

Figure 3. In the saltation-abrasion (S-A) model, the dependence of incision rate on channel gradientvaries with drainage area. Note that all calculations were made without applying the small-angleapproximation. Instead of plotting the data relative to sin(a), we plot it relative to a more intuitivevariable, channel gradient, tan(arcsin(sin(a))). The gradient at which incision rate falls below thebackground base level fall rate and permanent hanging valleys form, Shang, decreases with decreasingdrainage area. This reveals that small tributaries have a greater probability of creating hanging valleysthan large ones. Although difficult to perceive in this figure, the channel gradient at which the S-A modelachieves peak incision rates, Speak, increases with decreasing drainage area. Note that drainage basinswith drainage areas greater than �2.5 � 106 m2 will never produce permanent hangs from a response to afinite base level fall, indicating that the formation of permanent hangs in these circumstances will onlyoccur at relatively small drainage areas. Tributaries or drainage basins with drainage areas of greater than�2.5 � 106 m2 can only become permanently hung if the incision rate is permanently increased above themaximum incision rate allowable for that drainage area (thus plotting above a given line).


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ultimately increase the predicted values of both Speak andShang.

4.2. Instabilities in the Generalized Abrasion IncisionRule

[26] Unlike the saltation-abrasion incision rule, the gener-alized abrasion incision rule (adapted from Parker [2004])does not have a hump or discrete maximum value in therelation between incision rate and channel gradient (Figure 2).Although the exponent on S (sin(a)) in equation (12), iszero, there is still a functional dependence on the channelgradient through the Qt term defined in equation (3). In thegeneralized abrasion incision model, for a fixed sedimentflux, the incision rate increases asymptotically with channelgradient toward a maximum value, Imax (see Figure 2):

Imax ¼KGAbkwkbq

A1�bcUinitial: ð22Þ

This asymptotic approach to Imax prevents the incision ratefrom ever declining below the background rate of base levelfall in response to channel steepening. The only way thechannel could therefore create a permanent hanging valley(or waterfall) would be in the unlikely case where thesediment flux in the upper basin decreases to zero or if baselevel fall (or rock uplift) was maintained indefinitely at arate exceeding Imax.[27] Similar to the saltation-abrasion model, in response

to a finite period of increased base level fall temporaryhanging valleys will form in the generalized abrasionincision rule if the transient incision rate exceeds themaximum incision rate associated with the initial sedimentflux. Solving equation (22) for drainage area, we find themaximum drainage area at which a temporary hangingvalley could form

Atemp ¼kwk

bq

KGAbImax

Uinitial

! 11�bc

: ð23Þ

Following a pulse of incision, the oversteepened reach atAtemp decreases channel gradient because the main stemincision rate returns to just balancing the background rate ofbase level fall (Uinitial) while the tributary continues to inciseat a rate equal to Imax, resulting in a progressive decay ofhanging valley height and the maximum gradient of theoversteepened reach. In addition, sediment flux fromupstream increases in response to the incision, resulting ina positive feedback, accelerating incision of the upper lip ofthe oversteepened reach. This results in the eventual, ifasymptotic, readjustment of the tributary to graded condi-tions following the transient pulse of incision.[28] Evidence from the Waipaoa River and experimental

studies suggest that through most of the channel network,the base level fall signal experienced at a point along thechannel is not a discrete, on/off pulse of incision but rathergradually builds toward a maximum incision rate and thendeclines [Gardner, 1983; Crosby and Whipple, 2006;K. Berryman et al., The postglacial downcutting history inthe Waihuka tributary of the Waipaoa River, GisborneDistrict, New Zealand, and implications for tectonics andlandscape evolution, manuscript in preparation, 2007,

hereinafter referred to as Berryman et al., manuscript inpreparation, 2007]. This rise and fall in the wave of incisionexperienced at a tributary junction allows an initial signal ofincision to propagate up into the tributary before the largemagnitude incision rate potentially results in the formationof a hanging valley that effectively isolates the tributaryfrom the main stem. This partial communication of theinitial incision signal may increase tributary sediment fluxenough that when the large magnitude incision rate createsan oversteepened reach, the hanging tributary’s elevatedsediment flux is sufficient to allow eventual recovery.

5. Numerical Simulations of TransientLandscape Response

[29] We utilize the CHILD numerical landscape evolutionmodel to compare the transient responses of landscapesgoverned by four different stream incision rules [Tucker etal., 2001a; Tucker et al., 2001b; Gasparini et al., 2007]. TheCHILD numerical landscape evolution model provides anexplicit accounting of sediment production and transport atevery model node, thus offering an excellent tool forexploring the transient response of sediment flux–dependentchannel incision rules. Triggered by base level fall, channelincision accelerates sediment delivery from diffusion dom-inated hillslopes and generates a spatially and temporallycomplex sediment flux response. During each model run, atany point along the main stem channel the sediment fluxfluctuates as a consequence of both the local adjustment ofhillslopes and the integrated, upstream network response tothe pulse of incision. This unsteady sediment flux responsedirectly impacts the continued fluvial communication of thetransient base level fall signal through the system.[30] Although numerical landscape evolution models are

useful for studying the interaction between sediment pro-duction, sediment transport and channel incision at thenetwork scale, they also present limitations. In all modelruns, we define hillslope erosion only as a diffusive process,where the rate of sediment delivery is controlled by adiffusion coefficient, KD, and the local hillslope gradient.This limits the hillslope transport processes to creep andrain splash erosion. In our application of the CHILDnumerical model, we study the transient response withinlarge basins (1 � 108 m2) (Figure 1d) and thus use a largenode spacing of �100 m (Table 1). This large node spacing,and the subsequent numerical diffusion, contributes to theprogressive decay of the incision signal as it propagatesupstream. Although this influences evolution of the form ofthe transient signal in the stream power incision model(Figures 4e–4h), we are less concerned in the other models.Results from model runs that utilized smaller node spacingwere hard to distinguish from the runs at the larger nodespacing. As discussed by Whipple and Tucker [2002] andGasparini et al. [2006, 2007], large drainage area regions inmodels using sediment flux–dependent incision rulesrespond to pulses of incision largely in a transport-limitedmanner. Because this diminishes the impact of numericaldiffusion on the modeling of the sediment flux–dependentincision rules (our main focus), we are confident thatnumerical diffusion does not limit the findings of this study.[31] Another unintended consequence of large node spac-

ing is that the instantaneous base level fall at the outlet


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Figure 4. Numerical simulations of main stem profile evolution and normalized incision rates areobserved following base level fall. Normalized incision is calculated by dividing the incision rate by thelong-term, predisturbance background rate of base level fall. Shown are (a–d) plots that reflectsimulations governed by the transport-limited (T-L) incision rule and (e–h) plots that reflect simulationsgoverned by the stream power (S-P) incision rule. In Figures 4a and 4e, the gray line represents thechannel profile immediately after the base level fall; the dotted line represents the first profile illustratedafter the base level fall; and the dash-dotted line is the final profile illustrated. In Figures 4c and 4g, thedotted and dash-dotted lines illustrate the incision rate for the dotted and dash-dotted profiles in Figures 4aand 4e, respectively. In Figures 4b and 4f, gray lines reveal the evolution of the channel profile duringprogressive base level fall (no incision rates are shown for these times), and the dotted lines represent thefirst profile shown after the base level fall rate returns to the background rate; the dash-dotted line is thefinal profile shown. In Figure 4b, the profile changes very slowly, and therefore the differences betweeneach time plotted are not discernable (plotting times were chosen to show the pattern in incision rate).Note that the y axis differs between Figures 4c and 4d. In each of the plots, the time between each channelprofile and incision plot is not necessarily steady; rather, the times plotted were chosen to illustrate thegeneral behavior of the system. Note that the incision signal in the S-P case migrates up the main stem asa discrete step, while in the T-L case, the increase in high sediment delivery during the transient preventsthe downstream reaches from reequilibrating.


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creates a step whose channel gradient, instead of being avertical step, is the ratio between the vertical base level falland the horizontal node spacing. For example, an instanta-neous 100 m base level fall on a grid with 100 m nodespacing, instead of being near vertical, would have agradient close to 1 (45 degrees).[32] In order to test the response of each incision rule to

the base level fall scenarios discussed above, we first createfour steady state landscapes, each fully adjusted to aparticular incision rule. To do this, we start each modellandscape as a 10 km by 10 km square of random, low-amplitude topography with a single outlet in one corner andzero-flux edges. Each initial surface is then subjected touniform and steady rock uplift (or steady base level fall) atthe outlet until the network and hillslopes reach equilibrium.These four steady state landscapes provide the initialcondition for the base level fall experiments. As we onlystudy the basin’s response to finite base level fall, and not asustained change in the background uplift (or base level fall)rate (as explored by Gasparini et al. [2007]), all modelparameters are identical before and after the disturbance(Table 1).[33] For each incision rule, we first examine the responses

of the trunk stream to the two base level fall scenarios (asdefined in section 2; see Figures 4 and 5). Next, we focus onhow the trunk stream’s response influences tributary re-sponse (Figure 6) as a function of their size and positionwithin the drainage network. The modification of theincision signal as it propagates up the trunk stream provideseach tributary with a unique base level fall signal, poten-tially resulting in significant along-stream variation intributary response.

5.1. Transport-Limited Channel Incision Rule

[34] The transport-limited incision rule, although compu-tationally expensive in CHILD, is relatively robust over awide range of parameters (Table 1). Because the transport-limited incision rule is a nonlinear diffusion equation, theoversteepened reach created by the instantaneous base levelfall decays rapidly as the convex kink in the channel profilepropagates upstream (Figures 4a, 4b, and 6a). As a conse-quence, the highest incision rates of all four incision modelsare observed during the initial transport-limited adjustmentto instantaneous base level fall. As the base level fall signalpropagates upstream, the peak in incision rate movessurprisingly slowly upstream (Figures 4c and 4d). Althoughthe maximum incision rates are extremely high in thismodel, the location of the peak incision rate does not sweepupstream at the same rate as was observed in the othermodels. We suggest that this is a consequence of the lowerreaches being overwhelmed by the high sediment fluxesgenerated during transient adjustment.[35] For the scenario where base level lowers progres-

sively through time (Figures 4b and 4d, gray lines), much ofthe channel network efficiently steepens as the transport-limited channel responds to the temporarily higher rate ofbase level fall. No distinct knickpoints are created along thechannel profile (Figure 4b). Once the rate of base level fallreturns to the lower background value, a diffuse pulse ofincision sweeps up the main stem channel and slowlydecreases the channel gradient of the transiently oversteep-ened channels (Figures 4d and 6b). As in the instantaneous

base level fall example above, the highest incision ratesoccur near the outlet and progressively diminish as thesignal translates upstream.[36] The response of the landscape to the two base level

fall scenarios is very similar and it could be argued that theinstantaneous fall scenario, after a number of time steps,closely resembles the initial form of the progressive baselevel fall scenario. The only significant difference betweenthe two cases is that the response in the progressive baselevel fall model is slightly faster than the response to aninstantaneous base level fall. Neither of the two modelscreates temporary or permanent hanging valleys during thetransient response. The only delay in response is a conse-quence of the lag in hillslope sediment delivery followingthe pulse of incision.

5.2. Stream Power Incision Rule

[37] The detachment-limited stream power incision ruleruns much more efficiently than any of the other incisionrules in CHILD because sediment flux does not need to beexplicitly accounted for at every node. The CHILD model isrelatively robust to the parameters chosen for the streampower incision rule (Table 1). As discussed above andexpanded upon below, the only limitation of modeling thestream power incision model in CHILD is that numericaldiffusion rapidly attenuates the sharp breaks in channelgradient that should persist during the transient response(Figures 4e–4h).[38] For conditions where n = 1 and absent any numerical

diffusion, the transient response following instantaneousbase level fall should resemble a shock wave where theimposed step in the channel profile retreats upstream without changing form or magnitude. For values greater than orless than one, the form of the step is modified as it retreatsupstream [Weissel and Seidl, 1998; Tucker and Whipple,2002]. In the CHILD model, the step propagates upstream(Figure 4e) at a rate that is a power law function of drainagearea [Rosenbloom and Anderson, 1994], and slows stepwiseat tributary junctions but never creates permanent or tempo-rary hanging valleys (Figure 6c) [see also Niemann et al.,2001]. In the stream power model, the transient signalpropagates throughout the entire extent of the channel, reach-ing the headwaters without ever forming a hanging valley.[39] During progressive base level fall (Figures 4f, 4h,

and 6d), near the outlet, the channel steepens to a highergradient appropriate to the temporarily higher base level fallrate. Once the period of rapid base level fall ends, theoversteepened reach propagates upstream in the same man-ner as the discrete step did in the instantaneous base levelfall scenario. The oversteepened reach maintains the form ofa steady state channel segment adjusted to the higher rateof base level fall as it migrates upstream, so long as n = 1.Just as with the instantaneous base level fall, in the numericalsimulations the oversteepened reach is modified during itsupstream migration as a consequence of numerical diffusion.For tributaries, the differences between the instantaneousand prolonged base level fall responses are insignificant(Figures 6c and 6d).

5.3. Saltation-Abrasion Incision Rule

[40] Both the steady state form and transient response ofCHILD landscapes governed by the saltation-abrasion


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Figure 5. (a–h) Numerical simulations of main stem profile evolution and normalized incision rates areobserved following base level fall. Plots illustrate the initial transient response following disturbance, not thefull reestablishment of steady state channel form. Note that in each plot, the times between curves are notnecessarily equal. In the plots illustrating profile and incision rate response to an instantaneous base level fall(Figures 5a, 5c, 5e, and 5g), the gray lines indicate the profile immediately after the base level fall; the dottedlines indicate the first time step illustrated after the base level fall, and the dash-dotted lines indicate the lasttime step illustrated. In the plots illustrating the profile and incision rate response to progressive base levelfall (Figures 5b, 5d, 5f, and 5h), the gray lines show the evolving profile during progressive base level fall;the dotted lines indicate the first time step after the progressive base level fall; and the dash-dotted linesindicate the final time step illustrated. Figures 5a–5d reflect simulations governed by the simplifiedsaltation-abrasion (S-A) incision rule, while Figures 5e–5h reflect simulations governed by a generalizedabrasion (G-A) incision rule. In Figures 5a and 5c, instantaneous base level fall creates high-outlet gradients(between Speak and Shang) in the S-A model and initially retards the upstream transmission of the full incisionsignal. In Figures 5b and 5d, progressive base level fall in the S-A model creates an oversteepened channel(with gradients between Sss and Speak) that rapidly readjusts toward its predisturbance form. Figures 5e and5g demonstrate the rapid decay (at Imax) of the step created by instantaneous base level fall in the G-A model.In Figures 5f and 5h, the response following the disturbance is fast, but maximum incision rates are limitedby the maximum channel gradients created by the progressive base level fall.


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Figure 6. (a–h) Numerical simulations of tributary profile evolution are observed following base levelfall. In all plots, the dotted line indicates the profile after the first time step of adjustment, and the dash-dotted profile indicates the last time step illustrated. In Figures 6a and 6b, tributaries in the transport-limitedmodel adjust in concert with incision in the main stem, never developing hanging valleys. Figures 6c and 6ddemonstrate that in the stream power model, a wave of steepening and high incision rates sweeps up themain stem and through the tributaries. Tributaries initially steepen in proportion to the main stem as set bytheir drainage areas but then rapidly relax. Hanging valleys do not form. Figure 6e demonstrates that forinstantaneous base level fall in the saltation-abrasion model, the oversteepened reach locally diminishesthe incision rate at the outlet. This buffers the rapid upstream communication of the base level fall signal.Upstream tributaries consequently experience slow main stem incision rates and thus never createpermanent hanging valleys. Figures 6f, 6g, and 6h demonstrate the formation of temporary hangingtributary valleys as the pulse of incision rapidly migrates up the main stem. In these plots, the behavior ofthe tributary closest to the outlet, which has the smallest drainage area, is notably different from theevolution of the other tributaries. Note that in all of the plots, the times between curves are not necessarilyequal.


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incision rule are highly sensitive to the parameters used.Modeled landscapes with realistic channel gradients andnetwork densities are created only by a narrow combinationof values for hillslope diffusivity, Kt and KSA (Table 1). Thissensitivity is a direct consequence of the inclusion ofsaltation dynamics in the model (and thus the negativeexponents on drainage area and sin(a), equation (11)).[41] When subjected to a 50 m instantaneous base level

fall (Figure 5a), the discrete step created between the firstand second nodes of the main stem fails to retreat. As themodel runs forward in time, the step, locked in position atthe outlet, very slowly begins to decrease elevation. Foreach increment of lowering at the top of the step, the smallincision signal is rapidly transmitted through the upperportion of the basin (Figures 5c and 6e). The reason thestep fails to retreat and instead slowly lowers is because thechannel gradient created by the 50 m base level fall liesbetween Speak and Shang on the plot of incision rate versusgradient (Figures 2 and 3, solid lines). If the magnitude ofbase level fall were extreme, the resulting channel gradientat the outlet would exceed Shang (Figure 3). In this scenario,the incision rate in the oversteepened reach would dropbelow the background rate of base level fall and the entirenetwork would be permanently hung above the outlet (aresult confirmed in several simulations not reported here).For instantaneous base level falls significantly less than 50 m(given our 100 m cell size) the channel gradient at the outletis smaller and closer to Speak and consequently the stepdegrades faster than for larger base level falls. For largerbase level falls, the knickpoint remains at the outlet. Duringeach time step, a small fraction of the knickpoint’s height isable to propagate upstream. These incremental base levelfall signals are too small to create steep slopes or hangingvalleys in any tributary upstream (Figure 6e). We found itimpossible for an instantaneous base level fall event tocreate a temporary or permanent hanging tributary usingthe saltation-abrasion incision model, except at the basinoutlet itself. This reflects mostly the restricted size ofdrainage basins we can simulate with the moderate resolutionof 100 m cells. If we consider that the modeled basin isanalogous to a tributary within a much larger drainagebasin, an instantaneous base level fall at the outlet of themuch larger basin would create a progressive base level fallat the outlet of our simulated catchment; a scenario weconsider next.[42] For progressive base level fall (Figures 5b, 5d, and

6f), only moderate gradients develop at the outlet. Becausethese moderate values of sin(a) never exceed Speak, there isa rapid upstream propagation of the base level fall signalwithout ever creating a permanent or temporary hangingvalley at the basin outlet. If the prolonged main stem baselevel fall produces a value of sin(a) at the outlet that is closeto Speak, then the main stem channel incises at its maximumpotential rate, propagating the incision signal rapidlythrough the network. During progressive base level fall,rapid incision in the main stem can create tributary gradientsin excess of their local Shang or Speak, resulting in theformation of permanent or temporary hanging valleys(Figures 7 and 6f, respectively). For slower rates of pro-gressive base level fall, the value for sin(a) at the outlet issignificantly less than Speak and the transient signal prop-agates quickly through the network without forming hanging

tributaries except in the smallest of tributaries. In Figure 6fwe examine the response of the channel network after thebase level fall rate has returned to the background rate.Under this circumstance, the node spacing is too large andthe magnitude of base level fall is too small to createpermanent hanging valleys. All tributaries reequilibrate tothe new base level. It is important to note that for allprogressive base level fall scenarios, the main stem incisionsignal decreases in both amplitude and retreat rate as itpropagates upstream through the network (Figure 5d). Thismeans upper basin tributaries feel a smaller maximumincision rate for a longer duration. As discussed earlier,Shang, the gradient necessary to create a permanently hangingvalley decreases with decreasing drainage area. Althoughmost hanging valleys form lower in the catchment where themain stem incision rate is fast, (by creating channel gra-dients at junctions in excess of Shang), upper basin tributariesexperiencing relatively low main stem incision rates can stillform hanging valleys if their drainage area is sufficientlysmall. These findings are consistent with field observationsin New Zealand [Crosby and Whipple, 2006] and Taiwan[Wobus et al., 2006].[43] We can manipulate base level fall rates in the

saltation-abrasion model (or the generalized abrasion model)so that the incision rate in the main stem surpasses themaximum allowable incision rate in some or all of thetributaries. In Figure 7, we demonstrate that for differentmagnitudes of progressive base level (5�, 10� and 15�Uinitial), tributaries of different drainage area can becomehung above the main stem during progressive base level fall.The tributaries become elevated above the main stem whentheir junction is only capable of incision at a rate below thebackground rate of base level fall. Because the transientbase level fall signal is attenuated as it propagates upstream,we observe variations in response at different upstreamdistances, at different tributary drainage areas and fordifferent base level fall magnitudes.

5.4. Generalized Abrasion Incision Rule

[44] For all CHILD model runs using the generalizedabrasion incision rule, permanent hanging valleys neverform as a consequence of instantaneous or progressive baselevel fall, as expected from the analytical solutions discussedabove. Relative to model runs using the saltation-abrasionincision rule, landscapes predicted by the generalized abra-sion incision rule are much more stable over a wider rangeof parameter space (utilized values provided in Table 1). Inthe generalized abrasion incision model, the main stemresponse to both base level fall scenarios behaves similarlyto the channels governed by the transport-limited incisionrule (Figures 5e and 5f). The step created by the instanta-neous base level fall signal decays quickly as it retreatsupstream (Figure 5e). The plot of normalized incision rateshows the decrease in incision rate as the main stem signaldecreases gradient as it propagates upstream (Figure 5g).The tributaries (Figure 6g) contain temporary hangingvalleys that relax as the upstream end of the oversteepenedreach rounds off. By the end of the main wave of adjust-ment, the channel profile appears to have reached a steadyform but it is still steeper than the equilibrium gradient. Thisis a consequence of the excess sediment flux still beingdelivered off the adjusting hillslopes. Although the main


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stem channel may stabilize, it remains at disequilibriumuntil all portions (channels and hillslopes) of the landscapereturn to equilibrium. Contrasting the response to the pulseof incision, this return to equilibrium is a top-down processwhere the hillslopes have to adjust before the channelscomplete their adjustment.

[45] During the period of progressive, accelerated baselevel fall, the main stem channel rapidly adjusts by increas-ing channel gradient. When the initial base level fall rate isrestored (Figure 5f), the form of the profile resembles thatobserved halfway through the instantaneous base level fallresponse (Figure 5e). Because lower reaches of the mainstem are graded to the elevation of the outlet, the progres-sive base level fall scenario more rapidly adjusts backtoward steady state. This is reflected in a reduced basinresponse time and a smaller maximum incision rate(Figure 5h) and the failure to initiate permanent hangingtributaries (as discussed above with respect to Figure 6h).[46] Temporary hanging valleys do form around Atemp in

landscapes modeled in CHILD with the generalized abrasionincision rule as a consequence of the lag time in sediment fluxdelivery from tributary hillslopes (Figure 6g). Without suffi-cient sediment delivered from upstream, the incision rate atan oversteepened tributary junction decreases and a tempo-rary hanging valley forms. As the tributary’s hillslopesincrease sediment delivery in response to the initial partialcommunication of the incision signal upstream of the over-steepened junction, the incision rate in the hanging valleyincreases. Incision at the lip of the oversteepened reacheventually leads to the decay of the hanging valley andreestablishment of graded conditions.

6. Discussion

[47] Our analytical results and numerical simulationsdemonstrate that sediment flux–dependent incision rulespresent viable mechanisms for the formation of both per-manent and temporary hanging valleys. In landscapesgoverned by the two sediment flux–dependent incisionrules used here, the progressive modification of the baselevel fall signal as it propagates up the main stem deter-mines the magnitude and duration of the incision signal ateach tributary junction. Lower basin tributaries, located nearthe outlet where the base level fall initiates, feel the fastestincision rates for the shortest duration and thus have the

Figure 7. Formation of permanent hanging tributaries inthe saltation-abrasion model requires sufficiently fast mainstem incision rates. This figure examines tributary responsegiven three different rates of progressive base level fall.(a) For 5� the initial base level fall rate, only the twotributaries with the smallest drainage areas (the two closesto the outlet) form a hanging valley. (b) For 10� the initialbase level fall rate, all three of the smallest tributaries nearthe outlet form hanging valleys. (c) For 15� the initial baselevel fall rate, all tributaries in the basin become hangingvalleys. For each hanging tributary, the incision rate in themain stem outpaces the incision rate at the tributaryjunction. If the progressive base level fall is sustained, thehanging tributaries will increase in elevation indefinitely.Depending on the slopes of the oversteepened reaches(Sjunction), once the base level fall rate returns to thebackground rate, the tributaries either will remain aspermanent hanging valleys (if Sjunction > Shang) or willgradually reestablish grade with the main stem channel (ifSjunction < Shang).


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greatest potential to create large drainage area hangingvalleys. The pulse of incision is attenuated in a differentmanner for each incision model as it propagates upstream(Figures 4c, 4d, 4g, and 4h and 5c, 5d, 5g, and 5h), thusupper basin tributaries experience a long-duration signal at alower incision rate. This upstream attenuation of the mainstem incision signal plays an important role in determiningwhether particular tributaries become either temporary orpermanent hanging valleys. In the saltation-abrasion model,our modeling demonstrates that a permanent hanging valleyforms when the tributary channel gradient exceeds Shang.This threshold gradient decreases for lower drainage areatributaries (Figure 3). We also demonstrated that the gra-dients required to form temporary hanging valleys in boththe saltation-abrasion and the generalized abrasion incisionrules decrease at lower drainage areas. Consequently, fol-lowing a single pulse of incision, the resulting basin-widedistribution of permanent and temporary hanging valleys isgoverned by a complicated competition between four mainfactors: the magnitude of the initial base level fall at theoutlet; the rate of upstream attenuation of that incisionsignal as it propagates up the main stem; the lag time ofthe sediment flux response and the nonsystematic variationin tributary drainage areas within the channel network.[48] If the magnitude and duration of the incision signal

changes as it propagates upstream, the sediment flux–dependent incision rules suggest that tributaries of identicaldrainage area at different locations in the basin will not havethe same propensity to form hanging valleys. In a study ofhanging valleys in the Coast Ranges of Taiwan,Wobus et al.[2006] recognize that most hanging tributaries in their fieldarea have a trunk-to-tributary drainage area ratio greaterthan �10:1 [Wobus et al., 2006, Figure 11]. Lower in themain stem, our analysis supports the observation that mainstem drainage area must be significantly greater thantributary drainage area to create high enough tributarygradients to form hanging valleys. Our analysis also pro-vides evidence that this trunk-to-tributary drainage arearatio for hanging valley formation is sensitive to the fourfactors listed above, especially in the upper portion of thenetwork where the tributaries and the trunk stream may beof similar drainage area and both close to forming hangingvalleys.[49] Comparing the channel response to both instanta-

neous and progressive base level fall signals, we find thatextending the time of base level fall diminishes the magni-tude of the maximum incision rate and extends the durationof the pulse of incision. By distributing the same magnitudemain stem base level fall signal over a longer time period,tributary junctions have a greater probability of keepingpace with main stem incision and thus a lower likelihood ofcreating temporary or permanent hanging valleys. That said,we also recognize that in most field settings, large magni-tude instantaneous base level fall events will be significantlyless frequent than periods of progressive base level fall.[50] As anticipated, the transport-limited and detachment-

limited stream power incision rules did not generate eitherpermanent or temporary hanging valleys. In these modelruns, the base level fall signal was communicated all theway to the headwaters. In the transport-limited model, theresponse to base level fall was extremely rapid. The onlysignificant lag in channel response was due to overwhelm-

ing the channel by high initial sediment fluxes and the delayin hillslope adjustment to the incising streams. The modelbasin governed by the detachment limited incision rule didnot create hanging valleys at tributary junctions that grew inmagnitude or slowed the retreat rate any more than antic-ipated by the celerity model for detachment-limited knick-point retreat [Rosenbloom and Anderson, 1994; Whippleand Tucker, 1999; Crosby and Whipple, 2006].

6.1. Limitations and Recommendations forFuture Work

[51] In future analyses, it would be advantageous todecrease node spacing of the model to limit the effect ofnumerical diffusion on channel evolution and to allowtributary junctions to steepen above Shang. This could beachieved by running the code on a faster machine or on acluster or by simply budgeting longer run times for eachanalysis. Although the CHILD model offers an excellentenvironment for testing the behavior of sediment flux–dependent incision rules, we recognize that the treatment ofhillslope processes requires improvement. With moreexplicit hillslope erosion processes active, simulation ofthe rate and magnitude of sediment delivery to the channelsfollowing incision would be improved. We also suggest thatimproved parameterization of hillslope processes will allowhanging tributaries created by thresholds in the sedimentflux–dependent incision rules to continue to evolve andretreat upstream because of nonfluvial processes (such as:sapping, mass wasting, weathering, debris flows, etc.), asobserved at some field sites [e.g., Laity and Malin, 1985;Weissel and Seidl, 1997]. In addition, we have not addressedpotentially important adjustments in channel width, bedmaterial grain size, and hydraulic roughness.[52] Future analyses would also benefit from examining

the impact of spatially variable substrate erodibility on theattenuation or modification of pulses of incision and theretention or decay of hanging valleys. Adding this dimen-sion to the analysis would bring it closer to representing thefield sites we hope to better understand. Along these lines, itwould also be advantageous to consider more complicatedsignals of base level change. For example, how wouldmultiple base level fall events staggered in time influencethe sensitivity of tributaries to forming hanging valleys?The predisturbance gradient of a tributary channel is animportant factor in determining its sensitivity to a particularbase level fall signal. Variations in tributary gradient couldoccur as a consequence of local lithology in the tributary oras a consequence of incomplete adjustment to a previoustransient signal. If the tributary is already oversteepened(but not hanging) from a previous incision event, then itmight require a smaller main stem incision event to causethe tributary to hang. A well-adjusted tributary of similardrainage area, sediment flux and position in the network butlower channel gradient would not share the same sensitivityto the smaller magnitude incision event.

6.2. Implications for Landscape Evolution and FieldObservations

[53] Analytical and numerical investigations of theresponse of sediment flux–dependent incision rules to baselevel fall provide significant evidence for the formation ofoversteepened reaches at tributary junctions as a conse-


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quence of exceeding threshold gradients. Because modernfield observations provide an incomplete record of pastevents, the utilization of theoretical analyses, as presentedabove, provide an effective mechanism for testing sparsefield data against continuous modeled data. Our work in theWaipaoa River, New Zealand [Crosby and Whipple, 2006]and in the Coast Ranges of Taiwan [Wobus et al., 2006]provide useful data sets to compare with modeled behavior.[54] Numerous characteristics of the modeled behavior

align well with our observations from the Waipaoa Riverbasin [Crosby and Whipple, 2006]. In this fluvial catchment,a pulse of incision that initiated �18,000 years ago haslowered base level �80 m throughout the main channels inthe 2150 km2 basin on the North Island of New Zealand.Some �236 knickpoints are distributed on tributariesthroughout the basin, many of which are positioned atjunctions between tributaries and the trunk streams. Althoughthe cause of the pulse of incision is unclear, flights ofintermediate fluvial terraces suggest that trunk stream inci-sion was progressive rather than instantaneous. There is alsoevidence for a gradual rise and then decline in incision rate(Berryman et al., manuscript in preparation, 2007). Thisevidence for a rise and fall in incision rates fits well with themodeled behavior of sediment flux–dependent incisionrules because, even for an instantaneous base level fallsignal, the transport-limited nature of these incision rulesat large drainage provides a suitable mechanism to attenuatethe incision signal.[55] In the Waipaoa, large magnitude (50–100 m) knick-

points with well defined upstream lips are most frequentlyfound just upstream of tributary junctions, as predicted byboth sediment flux–dependent incision models. There areno observations of undercutting plunge pools or cap rockconditions that would facilitate the maintenance of a dis-crete kink at the knickpoint lip. Field observations suggestthat the minor retreat of the oversteepened reaches from thetributary junctions is dominated by nonfluvial processesincluding physical and chemical weathering, block failuresand other mass wasting processes. In both the WaipaoaRiver and Taiwan, we find that this retreat distance backfrom the main stem is drainage area–dependent [Crosbyand Whipple, 2006]. This suggests that though the processof retreat may be nonfluvial, it can still be sensitive to thegeometry of the drainage basin. Groundwater processes orthe combined contribution of fluvial and nonfluvial pro-cesses may contribute toward the retreat and decay ofhanging valleys.[56] In the Waipaoa, the tributary reaches upstream of the

knickpoints have two distinguishing characteristics: (1) thereis very little incision into either the bed of the hangingtributary or the knickpoint lip and (2) the hanging channelsare bare with almost no sediment on planar bedrock beds.The first observation aligns well with the saltation-abrasionmodel for the case where the oversteepened reach is greaterthan Shang. Alternatively, the observation could fit well witheither the saltation-abrasion model or the generalized abra-sion model given the assumption that sediment productionin the upper basins decreased dramatically and the hangingvalleys have not produced sufficient tools to erode either thechannel bed or the knickpoint lip. This is supported by theobservation that the hillslopes in many of the broad, flatbottomed hanging valleys only deliver sediment to their foot

slopes, not the channel. Extensive fill deposits border thehillslopes, storing the sediment derived from the hillslopesand inhibiting the transfer of material to the fluvial network.[57] Model simulations provide strong evidence that, for

channels governed by sediment flux–dependent incisionrules, two mechanisms extend the basin’s response timefollowing disturbance. First, because sediment deliveryfrom hillslopes appears to lag behind fluvial incision, thechannel’s return to equilibrium is prolonged as the hillslopesslowly reestablish equilibrium sediment delivery. The sec-ond mechanism that extends the response time is theformation of temporary or permanent hanging valleys inwhich potentially slower, lithology-dependent, nonfluvialprocesses may dominate the knickpoint retreat process.[58] Our simulations also suggest that in noncaprock

environments, the concept of a ‘‘knickpoint retreat rate’’needs to be carefully evaluated as the upstream communi-cation of a pulse of incision is neither continuous nor aprocess dominated by a single mechanism. Because therelative contribution of different erosion processes will varyas the signal propagates upstream, the measured retreat ratesin the main stem may not be able to be extrapolated into thetributaries. Multiple observations of transient features suchas terraces and knickpoints provide a basin-wide context tohelp differentiate between the different styles of signalpropagation. Even with this basin-wide context, it is diffi-cult to take locally measured retreat rates and scale themappropriately to constrain the rate of adjustment in basinscomplicated by network structure and nonuniform substrate.[59] Further analysis of how tributaries and trunk streams

interact will provide not only a better understanding of howrivers communicate signals of incision through their net-work of streams, but also shed light on the processes thatdictate the rate of sediment delivery to depositional basins.Because of the abrupt, large magnitude changes in impor-tant parameters such as channel gradient, water dischargeand sediment flux, the likelihood of process transitions attributary junctions complicate the transmission of thesesignals between trunk and tributary. The development ofhanging valleys not only impedes the upstream transmissionof subsequent incision signals and delays the equilibrationof sediment delivery from hillslopes, but also limits theconnectivity of pathways for mobile species such as fish.The loss of this connectivity can segregate and isolatebiological communities, potentially affecting species evolu-tion [e.g., Montgomery, 2000].

7. Conclusions

[60] We present evidence from both theoretical andnumerical analyses that sediment flux–dependent incisionrules predict the formation of temporary and permanenthanging valleys in fluvial networks responding to a finitepulse of incision. We find that four factors determine thedistribution of hanging valleys in fluvial networks: themagnitude of the pulse of incision; the rate of decay ofthe incision signal as it propagates upstream; the lag time ofthe sediment flux response and the drainage area of thetributary. The channel gradients required to create perma-nent or temporary hanging valleys decrease with decreasingdrainage area, thus making it possible for even upper basintributaries receiving an attenuated base level fall signal to


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become hung above the main stem. Although the general-ized abrasion incision rule provides mechanisms for theformation of temporary hanging tributaries, the formation ofpermanent hanging valleys in the saltation-abrasion incisionrule provides a better fit to our field observations of bothknickpoint form and process transitions at knickpoints. Wealso suggest that, though the saltation-abrasion incision ruleprovides a good fit to field data, any incision rule in whichthe incision rate begins to decline beyond some criticalchannel gradient will allow the formation of hangingvalleys. Therefore an important caveat to our findings isthat the formation of fluvial hanging valleys cannot at thisjuncture be exclusively associated with sediment flux–dependent incision models and thus do not unequivocallydemonstrate that these models are correct. Our findings dounequivocally demonstrate some important shortcomings ofthe standard detachment-limited and transport-limited riverincision models that will become important in some appli-cations. For instance, the development of hanging valleys intributaries extends the response times of landscapes wellbeyond those predicted by the stream power or transport-limited incision rules. An improved understanding of theerosion processes that modify the oversteepened reaches (orwaterfalls) once hanging valleys have formed will improveour predictions of landscape response time and the deliveryof sediment to depositional basins.

[61] Acknowledgments. The authors wish to thank reviewers NoahFinnegan, Gary Parker, Josh Roering, and Bob Anderson for insightfulreviews which considerably improved the clarity of an earlier version of thismanuscript. This work was supported through both an NSF grant (EAR-0208312 to KXW) and an NSF Graduate Research Fellowship (to B.T.C.).

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��B. T. Crosby, Department of Geosciences, Idaho State University,

Pocatello, ID 83209, USA. ([email protected])N. M. Gasparini, Department of Geology and Geophysics, Yale

University, New Haven, CT 06511, USA.K. X. Whipple, School of Earth and Space Exploration, Arizona State

University, Tempe, AZ 85287, USA.C. W. Wobus, Cooperative Institute for Research in Environmental

Sciences, University of Colorado, Boulder, CO 80309, USA.


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